32 ANALYTICAL MECHANICS
quite different from the simple rotation by which we accomplished the passage of the body from one of these positions to the other. But the result, which we have just obtained, is true not only for positions which are separated by finite distances but also for positions which are infinitely near each other. Therefore by giving the body infinitesimal rotational displacements about properly chosen axes it may be made to assume all the positions which it occupies during its actual motion.
36. Instantaneous Axis. —As the body is made to occupy the various positions of its actual motion the axis of rotation moves at right angles to itself and generates u cylinder whose elements are perpendicular to the guide plane. The elements of the cylinder are called instantaneous axes, because each acts as the axis of rotation at the instant when the body occupies a certain position. The curve of intersection of the cylinder and the guide plane is called the centrode.
The motion of a cylinder which rolls in a larger cylinder is a simple example of uniplanar motion. In this case t he common element of contact is the instantaneous axis. As the cylinder rolls different elements of the fixed cylinder become the axis of rotation.
Motion of translation and motion of rotation are special cases of uniplanar motion. In motion of translation the axis of rotation is infinitely far from the moving body. In rotation the cylinder formed by the instantaneous axes reduces to a Single line, i.e., the axis of rotation.
37. Theorem II. — Rotation about any axis is equivalent to a rotation through the same angle about a parallel axis and a translation in a direction perpendicular to it
The truth of this theorem will be seen from Fig. 23, where the rigid body A is brought from the position A to the position A' by a single rotation about an axis through the point 0 perpendicular to the plane of the paper. This displace-